How to find integrals of parent functions without any horizontal/vertical shift?

[PeaceMartian]

TL;DR Summary: How to find integrals of parent functions without any horizontal/vertical shift?

Say you were given the equation :

How would you find :

with a calculator that can only add, subtract, multiply, divide

Is there a general formula?

 

[Frabjous]

You have a function of the form Cx^p where p ≠ -1.
The integral is Cx^p+1/(p+1)

 

[pasmith]

The difficulty here is to compute (adequate approximations to) 9^{1/7} and 14^{8/7} = 14 \cdot 14^{1/7} using "a calculator that can only add, subtract, multiply, divide".

 

[Frabjous]

The difficulty here is to compute (adequate approximations to) 9^{1/7} and 14^{8/7} = 14 \cdot 14^{1/7} using "a calculator that can only add, subtract, multiply, divide".

126^1/7≈128^1/7=2
One could then do the polynomial expansion (2-x)⁷=126 and throw away higher terms to find corrections. Or do some iteration.